Problem: $\dfrac{ 5k - l }{ 2 } = \dfrac{ 6k + 8m }{ -8 }$ Solve for $k$.
Answer: Multiply both sides by the left denominator. $\dfrac{ 5k - l }{ {2} } = \dfrac{ 6k + 8m }{ -8 }$ ${2} \cdot \dfrac{ 5k - l }{ {2} } = {2} \cdot \dfrac{ 6k + 8m }{ -8 }$ $5k - l = {2} \cdot \dfrac { 6k + 8m }{ -8 }$ Multiply both sides by the right denominator. $5k - l = 2 \cdot \dfrac{ 6k + 8m }{ -{8} }$ $-{8} \cdot \left( 5k - l \right) = -{8} \cdot 2 \cdot \dfrac{ 6k + 8m }{ -{8} }$ $-{8} \cdot \left( 5k - l \right) = 2 \cdot \left( 6k + 8m \right)$ Distribute both sides $-{8} \cdot \left( 5k - l \right) = {2} \cdot \left( 6k + 8m \right)$ $-{40}k + {8}l = {12}k + {16}m$ Combine $k$ terms on the left. $-{40k} + 8l = {12k} + 16m$ $-{52k} + 8l = 16m$ Move the $l$ term to the right. $-52k + {8l} = 16m$ $-52k = 16m - {8l}$ Isolate $k$ by dividing both sides by its coefficient. $-{52}k = 16m - 8l$ $k = \dfrac{ 16m - 8l }{ -{52} }$ All of these terms are divisible by $4$ Divide by the common factor and swap signs so the denominator isn't negative. $k = \dfrac{ -{4}m + {2}l }{ {13} }$